Problem: Simplify and expand the following expression: $ \dfrac{5}{z + 1}+ \dfrac{3}{z - 6}+ \dfrac{2z}{z^2 - 5z - 6} $
Solution: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor the quadratic in the third term: $ \dfrac{2z}{z^2 - 5z - 6} = \dfrac{2z}{(z + 1)(z - 6)}$ Now we have: $ \dfrac{5}{z + 1}+ \dfrac{3}{z - 6}+ \dfrac{2z}{(z + 1)(z - 6)} $ The least common multiple of the denominators is: $ (z + 1)(z - 6)$ In order to get the first term over $(z + 1)(z - 6)$ , multiply by $\dfrac{z - 6}{z - 6}$ $ \dfrac{5}{z + 1} \times \dfrac{z - 6}{z - 6} = \dfrac{5(z - 6)}{(z + 1)(z - 6)} $ In order to get the second term over $(z + 1)(z - 6)$ , multiply by $\dfrac{z + 1}{z + 1}$ $ \dfrac{3}{z - 6} \times \dfrac{z + 1}{z + 1} = \dfrac{3(z + 1)}{(z + 1)(z - 6)} $ Now we have: $ \dfrac{5(z - 6)}{(z + 1)(z - 6)} + \dfrac{3(z + 1)}{(z + 1)(z - 6)} + \dfrac{2z}{(z + 1)(z - 6)} $ $ = \dfrac{ 5(z - 6) + 3(z + 1) + 2z} {(z + 1)(z - 6)} $ Expand: $ = \dfrac{5z - 30 + 3z + 3 + 2z}{z^2 - 5z - 6} $ $ = \dfrac{10z - 27}{z^2 - 5z - 6}$